Operators with Closed Range, Pseudo-Inverses, and Perturbation of Frames for a Subspace

Canad. Math. Bull. Vol. 42 (1), 1999 pp. 37 45 Operators with Closed Range, Pseudo-Inverses, and Perturbation of Frames for a Subspace Ole Christensen Abstract. Recent work of Ding and Huang shows that if we perturb a bounded operator (between Hilbert spaces) which has closed range, then the perturbed operator again has closed range. We extend this result by introducing a weaker perturbation condition, and our result is then used to prove a theorem about the stability of frames for a subspace. 1 Introduction Let H, K be Hilbert spaces. It is well known that every bounded operator T : K H with closed range has a generalized inverse, usually called the pseudo-inverse, orthemoore- Penrose inverse. In a recent paper Ding and Huang [DH2] find conditions implying that a perturbation of an operator with closed range again has closed range. They connect the results with norm estimates for the corresponding pseudo-inverse operators. In the first part of the present paper we extend one of Ding and Huang s results such that it fits with our applications in Section 3, which concerns perturbations of frames. A frame is a family { f i } of elements in a Hilbert space with the property that every element in the space can be written as a (infinite) linear combination of the frame elements. The question we consider is whether a family {g i } which is close to a frame { f i} is itself a frame. Beginning with work of Heil [H], there has been some interest in this question. Most of the work has been done with the goal to find results which are easy to apply to frames arising in wavelet theory, see e.g., [FZ], [GZ]. Together with Heil and Casazza, the present author has contributed with theoretical results [CC1], [C1], [C2], [CH]. From our point of view the major drawback of the results presented so far is that they can only be applied if {g i } is contained in span{ f i}.thatis,if{f i} is only a frame for span{ f i} and not for the underlying Hilbert space, the theory can not be applied without restrictions on {g i }. This is a problem, e.g., in sampling theory, where sequences { f ( λ i)} consisting of translates of the single function f L 2 (R) plays an important role. A sequence { f ( λ i )} canbeaframeforspan{ f ( λ i)} (see the papers [BL], [CCK] for sufficient conditions), but as shown in [CDH] it can not be a frame for L 2 (R). Easy examples suggest that such a case would be outside the scope of the theory. It is therefore a surprise for the author that a different way of proof (involving the above mentioned results on pseudo-inverse operators) leads to a perturbation theorem which is very similar to previous results, but which covers the case of a frame for a subspace. Received by the editors May 27, 1997; revised June 8, 1998. AMS subject classification: 42C15. c Canadian Mathematical Society 1999. 37

38 Ole Christensen The results presented here lead immediately to an extension of the celebrated Kadec 1/4 Theorem. 2 Operators with Closed Range Let K, H denote Hilbert spaces and B(K, H) the set of bounded linear operators from K into H. The range and the kernel of T B(K, H) will be denoted by R T and N T, respectively. Suppose that the operator T B(K, H) has closed range. The pseudo-inverse of T is, by definition, the uniquely determined operator T : H K satisfying: T Tx = x, x N T and T y = 0, y R T. It is well known [K, p. 231] that the operator T B(K, H) has closed range if and only if γ(t) := It can be shown [DH1] that if R T is closed, then inf Tx > 0. x NT, x =1 γ(t) = 1 T. Let V, W be subspaces of the same Hilbert space. If V 0, the gap between V and W is defined by: δ(v, W ):= sup dist(x, W ) = sup inf x y. x V, x =1 x V, x =1 y W Asaconventionweuseδ(0, W ) = 0. Usually δ is most conveniently found using the orthogonal projection P of H onto W : δ(v, W ) = δ(v, W ) = Given operators T, U B(K, H) welet δ N :=δ(n T,N U ). sup x Px. v V, x =1 Using this notation Ding and Huang [DH2, Theorem 3.1] have proven a stability result for the closedness of the range of an operator: Theorem 2.1 Let T, U B(K, H) and suppose that T has closed range. If δ 2 N + T U 2 T 2 < 1, thenr U is closed, and U T (1 δ N 2 ) 1/2 T U T. Theorem 2.1 can be reformulated by saying that if an operator T has closed range and U is a small perturbation of T (in the sense that T U < (1 δ N 2 ) 1/2 ), then also U has closed T

Operators with Closed Range, Frames 39 range. We need a more general stability result which will be the key to our most important frame result in the next section. Theorem 2.2 Let T, U B(K, H). Suppose that δ N < 1 and that there exist numbers λ 1 [0; 1[, λ 2 ] 1; [ and 0 such that (1) Tx Ux λ 1 Tx + λ 2 Ux + x, x K. Then (i) γ(u ) (1 λ 1)γ(T)(1 δ 2 N ) 1/2 (ii) If R T is closed and λ 1 + γ(t)(1 δ 2 N < 1, thenr ) 1/2 U is closed and Proof (i) For x K, U (1 + λ 2 ) T (1 λ 1 )(1 δ N 2 ) 1/2 T. Ux Tx Tx Ux (1 λ 1 ) Tx λ 2 Ux x, so and Ux (1 λ 1) Tx x, γ(u ) = inf{ Ux x N U, x =1} (1 λ 1)inf{ Tx x N U, x =1}. Now (i) follows by the calculation in [DH2, Lemma 3.4], where it is shown that inf{ Tx x NU, x =1} γ(t)(1 δ 2 N ) 1/2. (ii) R U is closed if γ(u ) > 0, and by (i) this is satisfied if λ 1 + < 1. Also, in this case U exists, and γ(t)(1 δ N 2 ) 1/2 U = 1 γ(u ) 1 (1 λ 1 ) T (1 δ N 2 ) 1/2 = (1 + λ 2 ) T (1 λ 1 )(1 δ 2 N ) 1/2 T. Remark From the point of view that U is considered as a perturbation of T it might be surprising that λ 2 can be arbitrarily large. But in a special case it is necessary to restrict λ 2, namely if U is not a priori known to be bounded. In this case the boundedness of U follows from the inequality (1) if we assume that λ 2 < 1. We need this observation in Theorem 3.2. In concrete cases it can be difficult to estimate δ N. Therefore it is important to notice that this number can be avoided in some special cases. Recall that an operator T B(K, H) is said to have an index if dim(n T ) < or if dim(h/r T ) <. In this case the index is defined as ind(t) = dim(n T ) dim(h/r T ).

40 Ole Christensen We shall only work with operators T which have closed range, in which case dim(h/r T ) = codim(r T ). In our terminology [G, Theorem V.3.6] now reads: Theorem 2.3 Let T, U B(K, H). Suppose that T has closed range and that T has an index. If there exist λ, 0 such that λ + γ(t) < 1 and Tx Ux λ Tx + x, x K, then U has closed range. Furthermore dim(n U ) dim(n T ), codim(r U ) codim(r T ), and ind(u ) = ind(t). 3 Applications to Frames In this section H denotes a separable Hilbert space with the inner product, linear in the first entry. We begin with some definitions. Afamily{f i } His a frame (for H) if A, B > 0: A f 2 f, f i 2 B f 2, f H. A and B are called frame bounds. { f i } H is a frame sequence if { f i} is a frame for span{ f i}. { f i } H is a Bessel sequence if at least the upper frame bound B exists. In this case one can define a bounded operator by T : l 2 (N) H, T{c i } := c i f i. T is usually called the pre-frame operator. Composing T with its adjoint operator T gives the frame operator S: H H, Sf = f, f i f i. If both of the frame conditions are satisfied, then S is invertible and self-adjoint, which immediately leads to the frame decomposition f = SS 1 f = f, S 1 f i f i, f H. The possibility of making such a decomposition of every f H isthereasonforthe importance of frames. For a more detailed discussion we refer to [HW]. Our goal here is to arrive at a perturbation theorem for frame sequences. Before we state our main result we need a lemma, which is proven in [C3]:

Operators with Closed Range, Frames 41 Lemma 3.1 (i) A Bessel sequence { f i } is a frame sequence if and only if R T is closed. (ii) A Bessel sequence { f i } is a frame for H if and only if R T = H. (iii) If { f i } is a frame, then the optimal bounds (i.e., maximal lower bound, minimal upper bound) are A = 1,B= T 2. T 2 To Bessel sequences { f i }, {g i} H we associate the pre-frame operators T, U. Corresponding to those operators we use the notation δ N from Section 2. The use of pseudo-inverse operators leads to a surprisingly simple argument for the lower frame bound in our main theorem: Theorem 3.2 Let { f i } H be a frame sequence with bounds A, B. Let {g i} H and suppose that there exists numbers λ 2, [0; 1[ and λ 1, 0such that (2) ( c i ( f i g i ) λ 1 c i f i + λ2 c i g i + c i 2) 1/2 for all scalars c 1,...,c n (n N). Then {g i } is a Bessel sequence with upper bound B ( 1+ λ + B ) 2. 1 λ 2 If furthermore δn < 1 and λ 1 + A(1 δn < 1, then{g 2 ) 1/2 i } is a frame sequence with lower bound A(1 δ 2 N ) ( 1 λ + A(1 δn 2 ) 1/2 ) 2. Proof { f i } is a Bessel sequence, so we can define a bounded linear operator T : l 2 (N) H, T{c i } = c i f i. Furthermore T Band R T is closed. The condition (2) implies that so c i g i c i ( f i g i ) + c i f i ( (1 + λ 1 ) c i f i + λ2 c i g i + c i 2) 1/2, {ci } n, 1+λ 1 + ( c i g i c i f i c i 2) 1/2, {ci } n 1 λ 2 1 λ. 2 A Cauchy sequence argument now shows that c ig i actually converges for all {c i } l 2 (N), and in (2) and the above estimates the finite sequences {c i } n can be replaced by {c i } l2 (N). If we define an operator U : l 2 (N) H, U{c i } = c i g i,

42 Ole Christensen we have T{c i } U{c i } λ 1 T{c i } +λ 2 U{c i } + {c i }, {c i } l 2 (N) and U{c i } 1+λ 1 T{c i } 1 λ + {c i } 2 1 λ 2 (1 + λ 1) B + 1 λ 2 {c i }, {c i } l 2 (N). This estimate shows that {g i } is a Bessel sequence with the upper bound ( (1 + λ1 ) B + 1 λ 2 )2 ( = B 1+ λ + ) 2 B. 1 λ 2 Now we assume that δ N < 1andλ 1 + 1 A(1 δn < 1. Since γ(t) = 2 ) 1/2 T A we have that λ 1 + γ(t)(1 δ 2 N < 1, so by Theorem 2.2, R ) 1/2 U is closed. Therefore {g i } is a frame 1 sequence by Lemma 3.1. The optimal lower bound is = γ(u ) 2,and U 2 γ(u) 2 ( (1 λ 1 ) ) 2 A(1 δ 2 N ) 1/2 ( =A(1 δ 2 N ) 1 λ 1 + λ 2 + A(1 δn 2 ) 1/2 ) 2. Remarks 1) The condition λ 2 < 1 is only used in the proof of the existence of the upper bound, so if we know that {g i } is a Bessel sequence we can remove this assumption (but our estimate for the upper bound is no more valid then). 2) It is possible to show a similar result with δ N replaced by the gap between the ranges of the operators U, T. However, the proof is more involved, cf. [CFL]. Easy examples show the optimality of the bounds on λ 1,λ 2,in Theorem 3.2: the conclusion fails if (2) is only satisfied with λ 1 = 1(orλ 2 =1or= A,δ N = 0). If { f i } is a frame for H, it is known that Theorem 3.2 holds with δ N replaced by 0 and that {g i } is a frame for H, cf. [CC1]. This leads trivially to an extension of the celebrated Kadec s 1/4 Theorem; observe that we denote the index by n and that i denotes the complex unit number in the following result: Proposition 3.3 Let {λ n } n=1, { n } n=1 R. Suppose that {e iλnx } n=1 is a frame for L 2 ( π, π) with bounds A, B. If there exists a constant L < 1/4 such that n λ n L and 1 cos πl +sinπl< A B,

Operators with Closed Range, Frames 43 then {e i nx } n=1 isaframeforl 2 ( π, π) with bounds ( A 1 ) 2 B (1 cos πl +sinπl),b(2 cos πl +sinπl) 2. A Proof This is just a trivial adjustment of the standard proof of Kadec s 1/4 Theorem. Observing that c n e iλ n( ) 2 B c n 2 for all finite sequences {c n }, the estimates from [Y, p. 42] gives that c n (e iλn( ) e i n( ) ) B(1 cos πl +sinπl)( cn 2) 1/2 ( A cn 2) 1/2. A similar version could have been written down using Theorem 3.2, assuming only that {e iλ nx } n=1 is a frame sequence. The version presented here is closely related to [DS, Lemma 3], which gives the same conclusion if L < ln( A 4B(e 1) +1) 1/2 however without estimates of the frame bounds. For A = B this condition is L < 0.1173, where we have the usual Kadec condition L < 1/4. Proposition 3.3 was independently and simultaneously observed by Balan [B]. In the general situation described in Theorem 3.2, the assumption δ N < 1 is needed, as demonstrated by the following example: Example 3.4 Let {e i } be an orthonormal basis for H. Then {f i} := {e 1,e 2,0, 0,0,...}is a frame sequence. Given ɛ>0, let {g i } = π { e 1, e 2, ɛ 3 e 3, ɛ 4 e 4,..., ɛ } n e n,...., An easy calculation shows that δ N = 1. By choosing ɛ small enough we can make {g i } as close to { f i } as we want, in the sense that c i ( f i g i ) ɛ ( c i 2) 1/2, {ci } n 3, but {g i } is not a frame sequence. As observed in Section 2, the introduction of δ N can be avoided if the operator T has an index. This condition has a nice interpretation in terms of the underlying frame sequence { f i } : dim(h/r T) < means that { f i } is a frame sequence for a space of finite codimension in H, and the case dim(n T ) < corresponds to what Holub [Ho] calls a near-riesz basis for span{ f i }, meaning that { f i} consists of a Riesz basis for this space plus finitely many elements. In the later case, Holub also shows that the excess, i.e., the

44 Ole Christensen number of elements which need to be deleted in order to obtain a Riesz basis for the space, is equal to dim(n T ). Theorem 3.5 Let { f i } be a frame sequence and suppose that the corresponding pre-frame operator T has an index. Let {g i } H and suppose that there exist numbers λ, 0 such that λ + γ(t) < 1 and ( c i ( f i g i ) λ c i f i + c i 2) 1/2 for all scalars c 1,...,c n (n N).Then{g i } is a frame sequence. The corresponding preframe operator U has an index, and dim(n U ) dim(n T ), codim(r U ) codim(r T ) and ind(u ) = ind(t). The part of Theorem 3.5 concerning the relation between various dimensions is particulary interesting in the case where T is a Fredholm operator, meaning that both dim(n T ) and codim(r T ) are finite. In this case Theorem 3.5 says that a perturbation can increase the dimension of the spanned space, but the excess will decrease with the same amount. This general result can be illustrated by an easy example in R 3 :Let{e i } 3 be an orthonormal basis for R 3 and let { { f i } 3 = {e 1, 0, 0}, {g i } 3 = e 1, 1 } 2 e 2, 0. { f i } 3 spans a one-dimensional subspace, and the excess is 2. {g i} 3 is a perturbation of { f i } 3 in the sense of Theorem 3.5, {g i} 3 spans a 2-dimensional subspace, and the excess is 1. For more results relating different perturbation conditions and excess, we refer to [CC2]. In concrete applications of the theorems discussed here it seems to be most convenient to find a value for such that n c i( f i g i ) ( n c i 2 ) 1/2 for all sequences {c i } n. This is what we did in Proposition 3.3, and this is also the key principle in [FZ]. In terms of the operators T, U this corresponds to estimating T U. Acknowledgements topic. The author wants to thank Peter G. Casazza for discussions on the References [B] R. Balan, Stability theorems for Fourier frames and wavelet Riesz bases. J. Fourier Anal. Appl. (5) 3(1997), 499 504. [BL] J. Benedetto and S. Li, The theory of multiresolution frames and applications to filter design. Appl. Comput. Harmon. Anal., to appear 1998. [CC1] P. G. Casazza and O. Christensen, Perturbation of operators and applications to frame theory. J. Fourier Anal.Appl.(5)3(1997), 543 557. [CC2], Frames containing a Riesz basis and preservation of this property under perturbations. SIAM J. Math. Anal. (1) 29(1998), 266 278. [CCK] P. G. Casazza, O. Christensen and N. Kalton, Frames of translates. Preprint. [C1] O. Christensen, Frame perturbations. Proc. Amer. Math. Soc. 123(1995), 1217 1220. [C2], A Paley-Wiener theorem for frames. Proc. Amer. Math. Soc. 123(1995), 2199 2202.

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